log(m)'s close to integers (sum convergence)
Don McDonald
parabola at paradise.net.nz
Mon Mar 17 13:41:36 CET 2003
In message <E18uiz2-0007hL-00 at mclean.mail.mindspring.net> you write:
Dear Seqfans,
Leroy Quet:
> I am afraid that this question might either be trivial or already
> answered, in a way, a little while back when this group was discussing
> this sequence.
>
>
> Is the sum of the reciprocals of all of the terms (in the above sequence)
> convergent?
>
> ie. Does sum{k=0 to oo} 1/a(k) converge?
>
> (There are many terms?? close to each other, so the sum may very well
> diverge.)
I very doubt that it would diverge.
It is easier than that. One term is ceil(e^n) the other is floor(e^n.)
the number of terms close to e^n is 0, 1, or 2.
>
> Thanks,
> Leroy Quet
>
It is convergent.
Comparison series is
2* (1/2 +1/4 +1/8 + 1/16+...)=2.
or a/(1-r) = 1/ (1-1/e.)
Therefore, I believe I created a similar sequence.
e^n is closer to an integer than any previous e^m. ..., 0 <= m <n.
A seqfan disputed my suggestion that e^n is easier to calculate
than many (all the intermediate) log(n).s.
Donald S. McDonald.
18.03.03 00:29
>
> Consider the sequence:
>
> >ID Number: A080021
> >Sequence: 2,3,7,20,148,403,1096,1097,2980,2981,8103,59874,162755,
> > 442413,1202604,3269017,8886110,8886111,24154952,24154953,
> > 65659969,178482301,3584912846,9744803446,26489122130,
> > 72004899337,195729609428
successive ratios of terms are order n:n+1 or 1:e^k.
I.e. duplicate geometric series.
> >Name: log(n) is closer to an integer than is log(m) for any m with
> > 2<=m<n.
> >Comments: Every term is floor(e^k)+r for some integers k and r with k>=1
> >and -1
> > <= r <= 1.
> >References Suggested by Leroy Quet (qqquet at mindspring.com), Jan 19 2003
> >Example: log(2) = 1-0.306..., log(3) = 1+0.0986..., log(7) = 2-0.0540...,
> > log(20) = 3-0.00426...
> >See also: Cf. A080022, A080023.
> >Keywords: nonn
> >Offset: 0
> >Author(s): Dean Hickerson (dean at math.ucdavis.edu), Jan 20 2003
> >Extension: More terms from Don Reble (djr at nk.ca), Jan 20 2003
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